Finite Group Databasebeta

← #14331274

Subgroups — #14331274

Order 1920.

848 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D5×#944C211yesmaximal, characteristic
2#11752449C211yesmaximal, characteristic
3C3×(C2×AGL(1,𝔽5))⋊Q8C211yesmaximal, characteristic
4#11751845C212yesmaximal
5#11751842C212yesmaximal
6#11751835C212yesmaximal
7#11751750C212yesmaximal
8#11751747C212yesmaximal
9#11751745C212yesmaximal
10#10617390—31nomaximal, Hall
11C2×C3×Q8×D5C2211yescharacteristic
12C2×D5×Dic6C2211yescharacteristic
13C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
14C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
15C2×C3×C4⋊AGL(1,𝔽5)C2211yescharacteristic
16C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
17C2×Dic3⋊AGL(1,𝔽5)C2212yes—
18C2×Dic3×AGL(1,𝔽5)C2212yes—
19#91833C411yescharacteristic, complemented
20C5×#944C411yescharacteristic, complemented
21D5×Dic3⋊C4C2212yes—
22D5×Dic3⋊C4C2212yes—
23C4×D5×Dic3C2211yescharacteristic
24#91389C411yescharacteristic, complemented
25#91385C411yescharacteristic, complemented
26#91327C412yescomplemented
27#91327C412yescomplemented
28#91235C2212yes—
29#91235C2212yes—
30C3×(C2×AGL(1,𝔽5))⋊C4C2212yes—
31C3×(C2×AGL(1,𝔽5))⋊C4C2212yes—
32(C2×AGL(1,𝔽5))⋊Dic3C2212yes—
33(C2×AGL(1,𝔽5))⋊Dic3C2212yes—
34(C2×AGL(1,𝔽5))⋊Dic3C2212yes—
35(C2×AGL(1,𝔽5))⋊Dic3C2212yes—
36(C2×AGL(1,𝔽5))⋊Dic3C2212yes—
37(C2×AGL(1,𝔽5))⋊Dic3C2212yes—
38#76497—51nomaximal, Hall
39D5×C4⋊Q8—31no—
40(C2×AGL(1,𝔽5))⋊Q8—32no—
41(C2×AGL(1,𝔽5))⋊Q8—31no—
42(C2×AGL(1,𝔽5))⋊Q8D311yescharacteristic
43#65711—32no—
44#65711—32no—
45#65711—32no—
46#65711—32no—
47#65690—32no—
48C22×ΓL(1,𝔽16)C2312yes—
49C22×ΓL(1,𝔽16)C2312yes—
50C22×C3×AGL(1,𝔽5)C2312yes—
51C22×C3×AGL(1,𝔽5)C2312yes—
52C2×C3×C5×Q8C2×C411yescharacteristic
53C2×Dic30C2×C411yescharacteristic
54C2×C5×Dic6C2×C411yescharacteristic
55C3×Q8×D5—41no—
56C3×Q8×D5—41no—
57C2×C3×C4×D5C2311yescharacteristic
58C2×C3×C4×D5C2311yescharacteristic
59C2×C3×C4×D5C2311yescharacteristic
60C2×C3×Dic10C2×C411yescharacteristic
61C2×C3×Dic10C2×C411yescharacteristic
62C2×C3×Dic10C2×C411yescharacteristic
63C2×#1204C2×C412yes—
64C2×D5×Dic3C2312yes—
65C2×D5×Dic3C2312yes—
66D5×Dic6—41no—
67D5×Dic6—41no—
68C4⋊ΓL(1,𝔽16)—41no—
69C4×ΓL(1,𝔽16)—41no—
70C3×C4⋊AGL(1,𝔽5)—41no—
71C3×C4×AGL(1,𝔽5)—41no—
72Dic3⋊AGL(1,𝔽5)—42no—
73Dic3×AGL(1,𝔽5)—42no—
74#6712C2×C412yes—
75#6712C2×C412yes—
76C4×Dic15C2×C411yescharacteristic
77C5×Dic3⋊C4C2×C412yes—
78C5×Dic3⋊C4C2×C412yes—
79C4×C5×Dic3C2×C411yescharacteristic
80Dic3⋊Dic5C2×C412yes—
81Dic3⋊Dic5C2×C412yes—
82#6669C2×C412yes—
83#6669C2×C412yes—
84Dic3×Dic5C2×C412yes—
85C2×#944—51no—
86C3×#391—51no—
87#5229—51no—
88#4978—52no—
89#4933—52no—
90#4655—52no—
91#4652—52no—
92#4651—52no—
93#4646—52no—
94C2×Q8×D5—31no—
95C2×Q8×D5C2×D311yescharacteristic
96C2×C4⋊AGL(1,𝔽5)—32no—
97C2×C4⋊AGL(1,𝔽5)—31no—
98C2×C4⋊AGL(1,𝔽5)C2×D311yescharacteristic
99C2×C4×AGL(1,𝔽5)—32no—
100C2×C4×AGL(1,𝔽5)—31no—
101C2×C4×AGL(1,𝔽5)C2×D311yescharacteristic
102C5×C4⋊Q8—31no—
103#4090—31no—
104#4090—31no—
105D5×C4⋊C4—32no—
106D5×C4⋊C4—32no—
107#4034—32no—
108#4034—32no—
109C42×D5—31no—
110#4015—31no—
111(C2×AGL(1,𝔽5))⋊C4—32no—
112(C2×AGL(1,𝔽5))⋊C4—32no—
113(C2×AGL(1,𝔽5))⋊C4—32no—
114(C2×AGL(1,𝔽5))⋊C4—32no—
115(C2×AGL(1,𝔽5))⋊C4—32no—
116(C2×AGL(1,𝔽5))⋊C4—32no—
117(C2×AGL(1,𝔽5))⋊C4—32no—
118(C2×AGL(1,𝔽5))⋊C4—32no—
119(C2×AGL(1,𝔽5))⋊C4C2×D312yes—
120(C2×AGL(1,𝔽5))⋊C4C2×D312yes—
121#2381—151noSylow
122C22×C3×D5C2411yescharacteristic
123C2×ΓL(1,𝔽16)—22no—
124C2×ΓL(1,𝔽16)—22no—
125C2×ΓL(1,𝔽16)—24no—
126C2×ΓL(1,𝔽16)—22no—
127C2×ΓL(1,𝔽16)—22no—
128C2×C3×AGL(1,𝔽5)—22no—
129C2×C3×AGL(1,𝔽5)—22no—
130C2×C3×AGL(1,𝔽5)—22no—
131C2×C3×AGL(1,𝔽5)—24no—
132C2×C3×AGL(1,𝔽5)—22no—
133C3×C5×Q8—41no—
134C2×C3×C4×C5C22×C411yescharacteristic
135C2×C3×C4×C5C22×C411yescharacteristic
136C2×C3×C4×C5C22×C411yescharacteristic
137C2×Dic15C22×C412yes—
138C2×Dic15C22×C412yes—
139Dic30—41no—
140C2×C5×Dic3C22×C412yes—
141C2×C5×Dic3C22×C412yes—
142C5×Dic6—41no—
143C2×C3×Dic5C22×C411yescharacteristic
144C2×C3×Dic5C22×C411yescharacteristic
145C2×C3×Dic5C22×C411yescharacteristic
146C3×C4×D5—21no—
147C3×C4×D5—21no—
148C3×C4×D5—21no—
149C3×C4×D5—21no—
150C3×C4×D5—21no—
151C3×C4×D5—21no—
152C3×Dic10—41no—
153C3×Dic10—41no—
154C3×Dic10—41no—
155#1204—42no—
156D5×Dic3—22no—
157D5×Dic3—22no—
158D5×Dic3—22no—
159D5×Dic3—22no—
160C22×C3×Q8—51no—
161C22×Dic6—51no—
162C2×C3×C4⋊C4—51no—
163C2×C3×C42—51no—
164#944—51no—
165#944—51no—
166#944—52no—
167#944—52no—
168#944—51no—
169#944AGL(1,𝔽5)11yescharacteristic, complemented
170C2×C4⋊Dic3—51no—
171C2×Dic3⋊C4—52no—
172C2×Dic3⋊C4—52no—
173C2×Dic3⋊C4—52no—
174C2×C4×Dic3—51no—
175C2×C4×Dic3—52no—
176C2×C4×Dic3—51no—
177C3×#95—52no—
178C3×#95—52no—
179C3⋊#95—52no—
180C3⋊#95—52no—
181C3⋊#95—52no—
182C3⋊#95—52no—
183C3⋊#95—52no—
184C3⋊#95—52no—
185C3⋊#95—52no—
186C3⋊#95—52no—
187C22×AGL(1,𝔽5)—32no—
188C22×AGL(1,𝔽5)—32no—
189C22×AGL(1,𝔽5)C22×D312yes—
190C22×AGL(1,𝔽5)C22×D312yes—
191C2×C5×Q8—31no—
192C2×C5×Q8C4×D311yescharacteristic
193Q8×D5—41no—
194Q8×D5—41no—
195Q8×D5—121no—
196Q8×D5—121no—
197C2×C4×D5—32no—
198C2×C4×D5—32no—
199C2×C4×D5C22×D311yescharacteristic
200C2×C4×D5C22×D311yescharacteristic
201C2×C4×D5C22×D311yescharacteristic
202C2×Dic10—31no—
203C2×Dic10—32no—
204C2×Dic10C4×D311yescharacteristic
205C2×Dic10C4×D311yescharacteristic
206C2×Dic10C4×D311yescharacteristic
207C4⋊AGL(1,𝔽5)—41no—
208C4⋊AGL(1,𝔽5)—121no—
209C4⋊AGL(1,𝔽5)—122no—
210C4×AGL(1,𝔽5)—41no—
211C4×AGL(1,𝔽5)—121no—
212C4×AGL(1,𝔽5)—122no—
213C5×C4⋊C4—32no—
214C5×C4⋊C4—32no—
215C42×C5—31no—
216C4⋊Dic5—32no—
217C4⋊Dic5—32no—
218#683—32no—
219#683—32no—
220#683—32no—
221#683—32no—
222C4×Dic5—31no—
223C4×Dic5—32no—
224C2×C4⋊Q8—151no—
225#391—51no—
226#391—152no—
227#391—151no—
228#387—152no—
229#387—152no—
230#387—152no—
231#387—152no—
232#383—152no—
233C22×C3×C5C23×C411yescharacteristic
234C2×C3×D5H(5,𝔽2)11yescharacteristic
235C2×C3×D5H(5,𝔽2)11yescharacteristic
236C2×C3×D52−1+412yes—
237C2×C3×D52−1+412yes—
238ΓL(1,𝔽16)—42no—
239ΓL(1,𝔽16)—42no—
240C3×AGL(1,𝔽5)—42no—
241C3×AGL(1,𝔽5)—42no—
242C3×C4×C5—21no—
243C3×C4×C5—21no—
244C3×C4×C5—21no—
245Dic15—22no—
246Dic15—22no—
247C3×Dic5—21no—
248C3×Dic5—21no—
249C3×Dic5—21no—
250C5×Dic3—22no—
251C5×Dic3—22no—
252C2×C3×Q8—51no—
253C2×C3×Q8—51no—
254C2×C3×Q8—51no—
255C2×C3×Q8—201no—
256C2×C3×Q8—201no—
257C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
258C22×C3×C4—51no—
259C22×C3×C4—51no—
260C22×C3×C4—51no—
261C22×C3×C4—52no—
262C22×C3×C4—52no—
263C22×Dic3—52no—
264C22×Dic3—52no—
265C22×Dic3—52no—
266C22×Dic3—52no—
267C2×Dic6—51no—
268C2×Dic6—52no—
269C2×Dic6—201no—
270C2×Dic6—201no—
271C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
272C3×C4⋊C4—201no—
273C3×C42—201no—
274C4⋊Dic3—201no—
275Dic3⋊C4—52no—
276Dic3⋊C4—52no—
277Dic3⋊C4—52no—
278Dic3⋊C4—52no—
279Dic3⋊C4—52no—
280Dic3⋊C4—52no—
281Dic3⋊C4—202no—
282Dic3⋊C4C2×AGL(1,𝔽5)12yes—
283Dic3⋊C4C2×AGL(1,𝔽5)12yes—
284C4×Dic3—51no—
285C4×Dic3—52no—
286C4×Dic3—202no—
287C4×Dic3—201no—
288C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
289C22×D5C23×D311yescharacteristic
290C2×AGL(1,𝔽5)—22no—
291C2×AGL(1,𝔽5)—22no—
292C2×AGL(1,𝔽5)—22no—
293C2×AGL(1,𝔽5)—22no—
294C2×AGL(1,𝔽5)—24no—
295C2×AGL(1,𝔽5)—62no—
296C2×AGL(1,𝔽5)—62no—
297C2×AGL(1,𝔽5)—62no—
298C2×AGL(1,𝔽5)—64no—
299C2×AGL(1,𝔽5)—62no—
300C5×Q8—41no—
301C5×Q8—121no—
302C2×C4×C5—32no—
303C2×C4×C5—32no—
304C2×C4×C5C2×C4×D311yescharacteristic
305C2×C4×C5C2×C4×D311yescharacteristic
306C2×C4×C5C2×C4×D311yescharacteristic
307C2×Dic5—32no—
308C2×Dic5—32no—
309C2×Dic5C2×C4×D311yescharacteristic
310C2×Dic5C2×C4×D311yescharacteristic
311C2×Dic5C2×C4×D311yescharacteristic
312C4×D5—21no—
313C4×D5—21no—
314C4×D5—21no—
315C4×D5—21no—
316C4×D5—21no—
317C4×D5—21no—
318C4×D5—62no—
319C4×D5—62no—
320C4×D5—62no—
321C4×D5—62no—
322Dic10—41no—
323Dic10—41no—
324Dic10—41no—
325Dic10—122no—
326Dic10—121no—
327C22×Q8—51no—
328C22×Q8—151no—
329C4⋊Q8—31no—
330C4⋊Q8—151no—
331C4⋊Q8—152no—
332C4⋊Q8—152no—
333C4⋊Q8—151no—
334C4⋊Q8—151no—
335C2×C4⋊C4—51no—
336C2×C4⋊C4—152no—
337C2×C4⋊C4—152no—
338C2×C4⋊C4—152no—
339C2×C4⋊C4—151no—
340C2×C42—51no—
341C2×C42—151no—
342C2×C42—151no—
343C2×C42—152no—
344#95—52no—
345#95—52no—
346#95—152no—
347#95—152no—
348#95—152no—
349#95—152no—
350#95—152no—
351#95—152no—
352#95—152no—
353#95—152no—
354C2×C3×C5#51811yescharacteristic
355C2×C3×C5#51912yes—
356C3×D5#56312yes—
357C3×D5#56311yescharacteristic
358C3×D5#56311yescharacteristic
359C23×C3—51no—
360C3×Q8—41no—
361C3×Q8—201no—
362C3×Q8—201no—
363C3×Q8—201no—
364C2×C3×C4—51no—
365C2×C3×C4—51no—
366C2×C3×C4—51no—
367C2×C3×C4—101no—
368C2×C3×C4—101no—
369C2×C3×C4—101no—
370C2×C3×C4—101no—
371C2×C3×C4—101no—
372C2×C3×C4—101no—
373C2×C3×C4—102no—
374C2×C3×C4—102no—
375C2×C3×C4—102no—
376C2×C3×C4—104no—
377C2×C3×C4—102no—
378C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
379C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
380C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
381C2×Dic3—52no—
382C2×Dic3—52no—
383C2×Dic3—102no—
384C2×Dic3—102no—
385C2×Dic3—102no—
386C2×Dic3—102no—
387C2×Dic3—102no—
388C2×Dic3—102no—
389C2×Dic3—102no—
390C2×Dic3—104no—
391C2×Dic3—102no—
392C2×Dic3C22×AGL(1,𝔽5)12yes—
393C2×Dic3C22×AGL(1,𝔽5)12yes—
394Dic6—41no—
395Dic6—202no—
396Dic6—201no—
397C22×C5C22×C4×D311yescharacteristic
398C2×D5#100411yescharacteristic
399C2×D5#100411yescharacteristic
400C2×D5#100712yes—
401C2×D5#101012yes—
402AGL(1,𝔽5)—42no—
403AGL(1,𝔽5)—42no—
404AGL(1,𝔽5)—122no—
405AGL(1,𝔽5)—122no—
406C4×C5—21no—
407C4×C5—21no—
408C4×C5—21no—
409C4×C5—62no—
410C4×C5—62no—
411Dic5—21no—
412Dic5—21no—
413Dic5—21no—
414Dic5—62no—
415Dic5—62no—
416C2×Q8—31no—
417C2×Q8—51no—
418C2×Q8—51no—
419C2×Q8—51no—
420C2×Q8—152no—
421C2×Q8—151no—
422C2×Q8—201no—
423C2×Q8—201no—
424C2×Q8—601no—
425C2×Q8—601no—
426C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
427C22×C4—51no—
428C22×C4—51no—
429C22×C4—51no—
430C22×C4—52no—
431C22×C4—52no—
432C22×C4—152no—
433C22×C4—152no—
434C22×C4—152no—
435C22×C4—152no—
436C4⋊C4—32no—
437C4⋊C4—32no—
438C4⋊C4—152no—
439C4⋊C4—152no—
440C4⋊C4—152no—
441C4⋊C4—152no—
442C4⋊C4—152no—
443C4⋊C4—152no—
444C4⋊C4—201no—
445C4⋊C4—602no—
446C4⋊C4—601no—
447C42—31no—
448C42—152no—
449C42—151no—
450C42—201no—
451C42—601no—
452C42—602no—
453C3×C5#238111yescharacteristic, complemented, Hall
454C22×C3—51no—
455C22×C3—51no—
456C22×C3—52no—
457C22×C3—52no—
458C22×C3C23×AGL(1,𝔽5)11yescharacteristic
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—101no—
463C3×C4—101no—
464C3×C4—101no—
465C3×C4—202no—
466C3×C4—202no—
467Dic3—22no—
468Dic3—22no—
469Dic3—102no—
470Dic3—102no—
471Dic3—202no—
472Dic3—202no—
473C2×C5#548611yescharacteristic
474C2×C5#557212yes—
475D5#564612yes—
476D5#567211yescharacteristic
477D5#573511yescharacteristic
478C23—51no—
479Q8—41no—
480Q8—121no—
481Q8—201no—
482Q8—201no—
483Q8—201no—
484Q8—601no—
485Q8—602no—
486C2×C4—32no—
487C2×C4—32no—
488C2×C4—51no—
489C2×C4—51no—
490C2×C4—51no—
491C2×C4—101no—
492C2×C4—101no—
493C2×C4—101no—
494C2×C4—101no—
495C2×C4—101no—
496C2×C4—101no—
497C2×C4—102no—
498C2×C4—102no—
499C2×C4—102no—
500C2×C4—104no—
501C2×C4—102no—
502C2×C4—152no—
503C2×C4—152no—
504C2×C4—302no—
505C2×C4—302no—
506C2×C4—302no—
507C2×C4—302no—
508C2×C4—302no—
509C2×C4—302no—
510C2×C4—302no—
511C2×C4—302no—
512C2×C4—304no—
513C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
514C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
515C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
516C2×C3—51no—
517C2×C3—52no—
518C2×C3—51no—
519C2×C3#6625811yescharacteristic
520C2×C3#6626212yes—
521C5#7649711yescharacteristic, complemented, Sylow
522C22—52no—
523C22—52no—
524C22—51no—
525C22—51no—
526C22C22×D3×AGL(1,𝔽5)11yescharacteristic
527C4—21no—
528C4—21no—
529C4—21no—
530C4—62no—
531C4—62no—
532C4—101no—
533C4—101no—
534C4—101no—
535C4—202no—
536C4—202no—
537C4—302no—
538C4—302no—
539C4—602no—
540C4—602no—
541C3#1061739011yescharacteristic, complemented, Sylow
542C2—51no—
543C2—52no—
544C2—51no—
545C2#1175684712yes—
546C2#1175687111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC2×C3×C5×Q8C2×C4
prime radical, cosocleC22×C3×D5C24
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5#2381
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
19. #91833C4semidirect
20. C5×#944C4semidirect
24. #91389C4semidirect
25. #91385C4semidirect
26. #91327C4semidirect
27. #91327C4semidirect
169. #944AGL(1,𝔽5)semidirect
453. C3×C5#2381semidirect
521. C5#76497semidirect
541. C3#10617390semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
D5×#9442
#117524492
C3×(C2×AGL(1,𝔽5))⋊Q82
#117518452
#117518422
#117518352
#117517502
#117517472
#117517452
#106173903
#764975